Despite the fact that this query may involve interventions that contradict the evidence, it is well-defined, as the intervention specifies a new submodel. Indeed, \( P(Y_{x}={y}^{\prime } \mid E = {e})\)  is given by [1]}
\( \sum _{{u}}P(Y_{{x}}({u})={y}^{\prime })P({u}|{e})\,.\) 
There are two main ways to resolve this type of questions; the Abduction-Action-Prediction paradigm and the Twin Network paradigm shown respectively in ML literature among others in [2]}, [3]}. In short given SCM \(M\)  with latent distribution \(P(U)\)  and evidence \({e}\) , the conditional probability \(P(Y_{{x}} \mid {e})\)  is evaluated as follows: 1) Abduction: Infer the posterior of the latent variables with evidence \({e}\)  to obtain \(P(U \mid {e})\) , 2) Action: Apply \(\text{do}({x})\)  to obtain submodel \(M_{x}\) , 3) Prediction: Compute the probability of \(Y\)  in the submodel \(M_{x}\)  with \(P(U \mid {e})\) . Meanwhile the Twin Network paradigm casts the resolution of counterfactual queries to Bayesian feed-forward inference by extending the SCM to represent both factual and counterfactual worlds at once [4]}.
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